An ancient dragon. Even his fires could not have harmed the One Ring, Gandalf tells Frodo in The Fellowship of the Ring Book 1, Chapter 2.
Asfaloth is Glorfindel's shining white horse. Glorfindel knows that Asfaloth will not let any rider fall off his back whom Glorfindel commands him to carry. It is Asfaloth who brings Frodo across the Ford of Bruinen, even when Frodo (under the influence of the Ringwraiths) is reluctant to go. Asfaloth is too fast even for the Riders; but even so, the ambush of the other four Ringwraiths seems certain to cut Frodo off before the Ford. Asfaloth runs right in front of the Ringwraith leader and across the Ford before they can reach Frodo. And he stands guard over Frodo's unconscious body until his friends can come and pick Frodo up.
Bill the Pony is basically the pet animal of the Fellowship, though he doesn’t get to go on their quest for very long. He appears in the novel in Bree, as the Hobbits are traveling to Rivendell. After the Hobbits' five ponies bolt from The Prancing Pony, the only pack animal left in the village to carry their luggage is a "bony, underfed, dispirited animal" (1.11.28) belonging to none other than the villain, Bill Ferny. The Hobbits are so desperate to keep traveling that they pay Ferny’s ridiculous asking price (luckily for Bill the Pony).
After a few days of travel with the Hobbits, even after the face-off with the Ringwraiths at Weathertop and Frodo's injury, "the poor beast had improved wonderfully; it already seemed fatter and stronger, and had begun to show an affection for its new masters, especially for Sam" (1.12.15). The fact that he's flourishing so much under such tough conditions tells us something about what life with Bill Ferny must have been like.
Bill is positively reborn in Rivendell: "he was glossy and seemed to have the vigour of youth" (2.3.71). Sam insists that Bill the Pony must accompany the Company; he says that Bill will starve himself if he is left behind. (We strongly doubt that this is the case, but it does seem to be true that Bill the Pony is very fond of Sam.)
As Bill sets out on the road, "he was the only member of the Company that did not seem depressed" (2.3.71). Bill the Pony proves once more the great power of Sam's love: Sam bursts into tears when Bill cannot follow the Fellowship into the Mines of Moria. Sam has huge affection, not only for other people, but also for animals and plants (after all, he is a gardener). This character trait is important considering how often Tolkien personifies the nature of Middle-earth itself. In a place where eagles can be transportation and horses and ponies can be personally loyal, it is essential to have a character who loves these creatures as they deserve.
Fatty Lumpkin (not to be confused with Fatty Bolger, Frodo’s Hobbit friend) is what Frodo and the Hobbits call Tom Bombadil's pony, a fat animal who ranges far and wide through the Old Forest. The Hobbits' ponies learn the smell of Fatty Lumpkin while Sam, Frodo, Pippin, and Merry are staying with Tom and Goldberry. So when the Hobbits fall under the power of the Barrow-wights, the ponies go and find safety with Fatty Lumpkin. (By the way, Fatty Lumpkin is a slang term for any fat animal or person).
Gwaihir is the fastest of the Great Eagles. When Radagast attempts to gather information about the Ring from his animal and bird friends, he instructs them to carry their information straight to Gandalf. Hence, Gwaihir flies to Gandalf's prison at Orthanc, where Saruman is keeping him trapped. Gwaihir agrees to carry Gandalf as far as Rohan, where Gandalf can find a horse. So Gwaihir saves the day; he feels a little bit forced as a plot device, since it's awfully convenient that a giant eagle just happens to come to Gandalf's prison when Gandalf most needs to escape. Still, Gwaihir's presence in The Fellowship of the Rings echoes a similar episode in The Hobbit with the Lord of the Eagles, so at least Tolkien is consistent.
Okay, we know this one isn't an animal, but hey, flora and fauna get along, right? Old Man Willow is the huge willow tree that enchants Merry, Frodo, and Pippin to sleep and then tries to swallow Merry and Pippin into its trunk. Not only do the Hobbits have to worry about the Ringwraiths and the Orcs, but now the very trees are trying to eat them as they travel.
Tom Bombadil comes along just in time to instruct the tree: "You should not be waking. Eat earth! Dig deep! Drink water! Go to sleep! Bombadil is talking!" (1.6.77). He pulls Merry and Pippin away from Old Man Willow. Later, Tom Bombadil comments, "Old grey Willow-man, he's a mighty singer; and it's hard for little folk to escape his cunning mazes. But Tom had an errand there, that he dared not hinder" (1.7.23).
The next day, Tom Bombadil tells the Hobbits tales of the Old Forest, in which Old Man Willow is featured. The Old Forest is filled with ancient trees that remember a time when they ruled the countryside, and their bitterness and anger has set them against intruders to the Forest. Old Man Willow is one of these. His "heart was rotten" but his "strength was green" (1.7.41) and he is filled with cunning. He draws power from the earth and influences the woods on both sides of the Withywindle.
Shadowfax is a horse of the Riddermark. Rohan is well-known for its horses, and Shadowfax is the best of them. Gandalf describes him: "By day his coat glistens like silver; and by night it is like a shade, and he passes unseen. Light is his footfall!" (2.1.184). Once Gandalf escapes Saruman's tower with the help of Gwaihir the Windlord, he tames Shadowfax and rides him from Rohan to the Shire, back to Bree, and then to Rivendell to meet Frodo when he arrives at Elrond's home.